A type I conjecture and boundary representations of hyperbolic groups

TL;DR

This paper explores boundary representations of hyperbolic groups, establishing their weak equivalence, characterizations of hyperbolic groups with cocompact amenable subgroups, and answering specific questions about groups acting on trees.

Contribution

It introduces new results on weak containment of representations, characterizes hyperbolic groups with certain properties, and addresses open questions in the context of groups acting on trees.

Findings

Any two boundary representations of a hyperbolic locally compact group are weakly equivalent.

Non-amenable hyperbolic groups with a cocompact amenable subgroup have homothetic proper length functions.

A type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup.

Abstract

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of \L. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.